Linear Algebra Examples

$[\begin{array}{c} 3 & - 4 \\ - 5 & 6 \end{array}] \cdot [\begin{array}{c} a \\ b \end{array}] = [\begin{array}{c} 5 \\ 3 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 3 & - 4 \\ - 5 & 6 \end{array}] [\begin{array}{c} a \\ b \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 3 a - 4 b \\ - 5 a + 6 b \end{array}] = [\begin{array}{c} 5 \\ 3 \end{array}]$

Step 2

Write as a linear system of equations.

$3 a - 4 b = 5$

$- 5 a + 6 b = 3$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $a$ in $3 a - 4 b = 5$ .

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Step 3.1.1

Add $4 b$ to both sides of the equation.

$3 a = 5 + 4 b$

$- 5 a + 6 b = 3$

Step 3.1.2

Divide each term in $3 a = 5 + 4 b$ by $3$ and simplify.

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Step 3.1.2.1

Divide each term in $3 a = 5 + 4 b$ by $3$ .

$\frac{3 a}{3} = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

Step 3.1.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- 5 a + 6 b = 3$

Step 3.2

Replace all occurrences of $a$ with $\frac{5}{3} + \frac{4 b}{3}$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $- 5 a + 6 b = 3$ with $\frac{5}{3} + \frac{4 b}{3}$ .

$- 5 (\frac{5}{3} + \frac{4 b}{3}) + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $- 5 (\frac{5}{3} + \frac{4 b}{3}) + 6 b$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$- 5 (\frac{5}{3}) - 5 \frac{4 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.2

Multiply $- 5 (\frac{5}{3})$ .

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Step 3.2.2.1.1.2.1

Combine $- 5$ and $\frac{5}{3}$ .

$\frac{- 5 \cdot 5}{3} - 5 \frac{4 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.2.2

Multiply $- 5$ by $5$ .

$\frac{- 25}{3} - 5 \frac{4 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$\frac{- 25}{3} - 5 \frac{4 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.3

Multiply $- 5 \frac{4 b}{3}$ .

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Step 3.2.2.1.1.3.1

Combine $- 5$ and $\frac{4 b}{3}$ .

$\frac{- 25}{3} + \frac{- 5 (4 b)}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.3.2

Multiply $4$ by $- 5$ .

$\frac{- 25}{3} + \frac{- 20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$\frac{- 25}{3} + \frac{- 20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.4

Simplify each term.

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Step 3.2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{25}{3} + \frac{- 20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{25}{3} - \frac{20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- \frac{25}{3} - \frac{20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- \frac{25}{3} - \frac{20 b}{3} + 6 b = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.2

To write $6 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{25}{3} - \frac{20 b}{3} + 6 b \cdot \frac{3}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.3

Combine $6 b$ and $\frac{3}{3}$ .

$- \frac{25}{3} - \frac{20 b}{3} + \frac{6 b \cdot 3}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.4

Combine the numerators over the common denominator.

$- \frac{25}{3} + \frac{- 20 b + 6 b \cdot 3}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 25 - 20 b + 6 b \cdot 3}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.6

Multiply $3$ by $6$ .

$\frac{- 25 - 20 b + 18 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.7

Add $- 20 b$ and $18 b$ .

$\frac{- 25 - 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.8

Rewrite $- 25$ as $- 1 (25)$ .

$\frac{- 1 \cdot 25 - 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.9

Factor $- 1$ out of $- 2 b$ .

$\frac{- 1 \cdot 25 - (2 b)}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.10

Factor $- 1$ out of $- 1 (25) - (2 b)$ .

$\frac{- 1 (25 + 2 b)}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.2.2.1.11

Move the negative in front of the fraction.

$- \frac{25 + 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- \frac{25 + 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- \frac{25 + 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$- \frac{25 + 2 b}{3} = 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3

Solve for $b$ in $- \frac{25 + 2 b}{3} = 3$ .

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Step 3.3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{25 + 2 b}{3}) = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2

Simplify both sides of the equation.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Simplify $- 3 (- \frac{25 + 2 b}{3})$ .

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Step 3.3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.1.1

Move the leading negative in $- \frac{25 + 2 b}{3}$ into the numerator.

$- 3 \frac{- (25 + 2 b)}{3} = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (25 + 2 b)}{3}) = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (25 + 2 b)}{3}) = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.1.1.1.4

Rewrite the expression.

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.1.1.2

Multiply.

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Step 3.3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (25 + 2 b) = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.1.1.2.2

Multiply $25 + 2 b$ by $1$ .

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 3 \cdot 3$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Multiply $- 3$ by $3$ .

$25 + 2 b = - 9$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 9$

$a = \frac{5}{3} + \frac{4 b}{3}$

$25 + 2 b = - 9$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.3

Move all terms not containing $b$ to the right side of the equation.

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Step 3.3.3.1

Subtract $25$ from both sides of the equation.

$2 b = - 9 - 25$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.3.2

Subtract $25$ from $- 9$ .

$2 b = - 34$

$a = \frac{5}{3} + \frac{4 b}{3}$

$2 b = - 34$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.4

Divide each term in $2 b = - 34$ by $2$ and simplify.

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Step 3.3.4.1

Divide each term in $2 b = - 34$ by $2$ .

$\frac{2 b}{2} = \frac{- 34}{2}$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $2$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{- 34}{2}$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.4.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 34}{2}$

$a = \frac{5}{3} + \frac{4 b}{3}$

$b = \frac{- 34}{2}$

$a = \frac{5}{3} + \frac{4 b}{3}$

$b = \frac{- 34}{2}$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Divide $- 34$ by $2$ .

$b = - 17$

$a = \frac{5}{3} + \frac{4 b}{3}$

$b = - 17$

$a = \frac{5}{3} + \frac{4 b}{3}$

$b = - 17$

$a = \frac{5}{3} + \frac{4 b}{3}$

$b = - 17$

$a = \frac{5}{3} + \frac{4 b}{3}$

Step 3.4

Replace all occurrences of $b$ with $- 17$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $a = \frac{5}{3} + \frac{4 b}{3}$ with $- 17$ .

$a = \frac{5}{3} + \frac{4 (- 17)}{3}$

$b = - 17$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{5}{3} + \frac{4 (- 17)}{3}$ .

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Step 3.4.2.1.1

Combine the numerators over the common denominator.

$a = \frac{5 + 4 (- 17)}{3}$

$b = - 17$

Step 3.4.2.1.2

Simplify the expression.

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Step 3.4.2.1.2.1

Multiply $4$ by $- 17$ .

$a = \frac{5 - 68}{3}$

$b = - 17$

Step 3.4.2.1.2.2

Subtract $68$ from $5$ .

$a = \frac{- 63}{3}$

$b = - 17$

Step 3.4.2.1.2.3

Divide $- 63$ by $3$ .

$a = - 21$

$b = - 17$

$a = - 21$

$b = - 17$

$a = - 21$

$b = - 17$

$a = - 21$

$b = - 17$

$a = - 21$

$b = - 17$

Step 3.5

List all of the solutions.

$a = - 21, b = - 17$